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Volume 37, Issue 3
Global Solutions of Modified One-Dimensional Schrödinger Equation

Ting Zhang

DOI: 10.4208/cmr.2021-0015

Commun. Math. Res., 37 (2021), pp. 350-386.

Published online: 2021-06

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  • Abstract

In this paper, we consider the modified one-dimensional Schrödinger equation:
$$(D_t-F(D))u=λ|u|^2u,$$

where F(ξ) is a second order constant coefficients classical elliptic symbol, and with smooth initial datum of size $ε≪1$. We prove that the solution is global-in-time, combining the vector fields method and a semiclassical analysis method introduced by Delort. Moreover, we get a one term asymptotic expansion for $u$ when $t→+∞$.

  • Keywords

Schrödinger equation, semiclassical Analysis, global solution.

  • AMS Subject Headings

Primary: 35Q55, Secondary: 35A01, 35B40, 35S50

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CMR-37-350, author = {Zhang , Ting}, title = {Global Solutions of Modified One-Dimensional Schrödinger Equation}, journal = {Communications in Mathematical Research }, year = {2021}, volume = {37}, number = {3}, pages = {350--386}, abstract = {

In this paper, we consider the modified one-dimensional Schrödinger equation:
$$(D_t-F(D))u=λ|u|^2u,$$

where F(ξ) is a second order constant coefficients classical elliptic symbol, and with smooth initial datum of size $ε≪1$. We prove that the solution is global-in-time, combining the vector fields method and a semiclassical analysis method introduced by Delort. Moreover, we get a one term asymptotic expansion for $u$ when $t→+∞$.

}, issn = {2707-8523}, doi = {https://doi.org/10.4208/cmr.2021-0015}, url = {http://global-sci.org/intro/article_detail/cmr/19265.html} }
TY - JOUR T1 - Global Solutions of Modified One-Dimensional Schrödinger Equation AU - Zhang , Ting JO - Communications in Mathematical Research VL - 3 SP - 350 EP - 386 PY - 2021 DA - 2021/06 SN - 37 DO - http://doi.org/10.4208/cmr.2021-0015 UR - https://global-sci.org/intro/article_detail/cmr/19265.html KW - Schrödinger equation, semiclassical Analysis, global solution. AB -

In this paper, we consider the modified one-dimensional Schrödinger equation:
$$(D_t-F(D))u=λ|u|^2u,$$

where F(ξ) is a second order constant coefficients classical elliptic symbol, and with smooth initial datum of size $ε≪1$. We prove that the solution is global-in-time, combining the vector fields method and a semiclassical analysis method introduced by Delort. Moreover, we get a one term asymptotic expansion for $u$ when $t→+∞$.

Zhang , Ting. (2021). Global Solutions of Modified One-Dimensional Schrödinger Equation. Communications in Mathematical Research . 37 (3). 350-386. doi:10.4208/cmr.2021-0015
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